Relationship between Foreign Exchange and Commodity

Volatilities using High-Frequency Data Derrick Hang

Economics 201 FS, Spring 2010

Academic honesty pledge that the assignment is in compliance with the Duke Community

Standard as expressed on pp. 5-7 of "Academic Integrity at Duke: A Guide for Teachers and

Undergraduates "

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1. Introduction

Understanding the factors that drive asset volatility, or risk, is critical to the overall

understanding of movements in the financial markets. Several financial applications, such as

those in derivative pricing and portfolio management, are all dependent on the ability to

accurately model and predict volatility. Recent advancements in the financial theory, driven by

the application of high-frequency data, have produced elegant models that are simpler to

implement and have better measurement and forecasting capabilities than their predecessors. The

Heterogeneous Autoregressive (HAR) model proposed by Corsi (2003), for instance, performs a

simple linear regression to obtain estimates for realized variance. In the financial literature,

several empirical studies exist that use these models to describe the activity in the foreign

exchange markets; however, relatively few apply this framework to analyze the relationship

between the volatility in foreign exchange with the volatility in commodities.

For the most part, the link between the foreign exchange and commodity markets is well-

documented. Cashin et al. (2004) finds evidence of a long-term connection between real

exchange rates and commodity prices for approximately one-third of the commodity-exporting

countries. Exchange rates that pair the currency of a heavily commodity-exporting country to the

currency of a heavily commodity-importing country are seen in the market to follow the price

movements of the underlying commodity. Chabin (2009) rationalizes this and similar findings by

suggesting increases in commodity prices be viewed as a terms of trade improvement that is

equivalent to a transfer of wealth from commodity-importing to commodity-exporting countries.

Turning the focus of currency-commodity analysis to volatility, Diebold and Yilman (2010) find

indications of volatility spillover from the US commodity to the US foreign exchange markets.

We posit that, in particular, commodity currencies, due to their strong relationship with the

commodity markets, are also more likely to exhibit a link in their volatilities; more specifically,

we test for whether the volatility in the commodity can be indicative of the volatility of its

respective commodity currency. We implement our analysis using both a simple linear

regression approach and a more complex Bayesian method to provide a check for consistency in

our results. In both of the models, we discover some evidence of commodity volatility to be

indicative of the volatilities strong commodity currencies in the first half of 2009.

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The rest of this paper is structured as follows: section 2 provides theoretical background behind

the volatility measures and regression models implemented in this analysis; section 3 introduces

the dataset of foreign exchange and commodity prices; section 4 discusses research methods;

section 5 presents the empirical results obtained from the regressions, and section 6 offers

conclusive remarks about the findings and suggestions for further research.

2. Background

2.1 Stochastic Model of Returns

Implicit in our calculations for realized variance in section 2.2, we assume the general stochastic

model for price evolution, which indicates that the logarithm of prices p(t) of an asset can be

defined by the following differential equation:

𝑑𝑝 𝑡 = 𝑢 𝑡 𝑑𝑡 + 𝜎 𝑡 𝑑𝑊 𝑡 + κ 𝑡 𝑑𝑞 𝑡 (2.1.1)

Where u(t) is the time varying drift component, σ(t)dW(t) is the time-varying volatility

component, Wt represents a standard Brownian motion, and σ(t) is the volatility level. κ(t)dq(t)

represents a jump component where κ(t) is the magnitude of the jump and q(t) is a counting

process. However, it is generally assumed that jumps occur infrequently.

2.2 Asset Return Volatility Models

The geometric returns rt ,j used in our model is obtained by taking the first lagged difference of

the logarithm of intraday prices pt,j, shown below. Applying this formula in a rolling fashion, we

acquire a series of intraday returns for each day t.

𝑟𝑡 ,𝑗 = 𝑝𝑡 ,𝑗 − 𝑝𝑡 ,𝑗−1 (2.2.1)

To obtain the values for the daily realized variance for our initial regressions, we took the sum of

the m squared intraday returns for each day t. This measure of realized variance (RV)

asymptotically approaches the integrated variance plus the jump component as the sampling

frequency approaches infinity.

𝑅𝑉𝑡 = 𝑟𝑡 ,𝑗2

𝑚

𝑗=1

→ 𝜎𝑠2𝑑𝑠

𝑡

𝑡−1

+ κ𝑠2

𝑡−1<𝑠≤𝑡

(2.2.2)

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In addition, we also considered the realized absolute value (RAV) measurement of volatility,

which is defined to be the sum of the absolute value of intraday returns for each day t. As the

sampling frequency approaches infinity, RAV asymptotically approaches the integrated

volatility.

𝑅𝐴𝑉𝑡 = 𝜋

2𝑚 |𝑟𝑡,𝑗 |

𝑚

𝑗=1

→ 𝜎𝑠𝑑𝑠

𝑡

𝑡−1

2.2.3

It should be noted that RV and RAV are defined in different units and, therefore, cannot be

compared directly.

2.3 Regression Models

In this paper, we are largely dependent upon two different approaches, the Heterogeneous

Autoregressive model and the Bayesian Dynamic Linear model, to perform our analysis.

2.3.1 Heterogeneous Autoregressive Models

The first approach utilizes the Heterogeneous Autoregressive (HAR) model developed in Corsi

(2003) for predicting realized variance. The HAR-RV model is a simple linear regression that is

able to capture the persistence or time trends in an underlying time series by including its

average lagged time series in the regression. Consider the following equation:

𝑅𝑉𝑡,𝑡+𝑕 =1

𝑕 𝑅𝑉𝑘

𝑡+𝑕

𝑘=𝑡+1

(2.3.1)

Where RVt,t+h is the average RV over the given time span h. For the one-day ahead forecasting of

RVt , we define a weekly regressor to be the average of the RVs of the past 5 days up to time t

and a monthly regressor is the average of RVs of the past 22 days up to time t. We can now

present the HAR-RV model to be

𝑅𝑉𝑡+1 = 𝛼 + 𝛽𝑑𝑅𝑉𝑡 + 𝛽𝑤𝑅𝑉𝑡−5,𝑡 + 𝛽𝑚𝑅𝑉𝑡−22,𝑡 + 𝜀𝑡+1 (2.3.2)

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Where RVt , RVt,t-5 , RVt,t-22 represent lagged daily, weekly and monthly realized variance, RVt+1

is the one-day ahead forecasted realized variance, and βd ,βw ,βm are the corresponding regression

coefficient.

Likewise, we also define a HAR-RAV model specified in the same fashion as the HAR-RV

model, using RAV in place of RV.

𝑅𝐴𝑉𝑡+1 = 𝛼 + 𝛽𝑑𝑅𝐴𝑉𝑡 + 𝛽𝑤𝑅𝐴𝑉𝑡−5,𝑡 + 𝛽𝑚𝑅𝐴𝑉𝑡−22,𝑡 + 𝜀𝑡+1 (2.3.3)

Ghysels and Forberg (2004) show that RAV regressors are more robust estimators of volatility

than the RV sum of square measurement. However, note that the specific coefficient estimates

obtained from the HAR-RAV model cannot be compared to those obtained from the HAR-RV

model; in any case, it is still worthwhile to explore within-model significance to corroborate the

overall findings.

2.3.2 Bayesian Models

For our second approach, we chose to implement a univariate Bayesian dynamic linear model in

place of the simple HAR models. The rationale behind this model choice is two-fold: the

Bayesian framework provides a nice contrast to the previous analysis from which we can

compare results to check for consistency; the Bayesian dynamic model allows for time-varying

coefficient for the regressors, which provides another check for coefficient significance

throughout the sample period.

The full specifications and rationale behind the Bayesian dynamic linear model (DLM) can be

found in West and Harrison (1997). Due to the complex nature of Bayesian modeling, we only

include the descriptions of the major updating equations and assumptions in this paper. The

general DLM model equations specified in our regression is as follows:

𝑦𝑡 = 𝛼𝑡 + 𝛽1,𝑡𝑦𝑡−1 + 𝛽2,𝑡𝑥𝑡−1 + 𝑣𝑡 𝑤𝑕𝑒𝑟𝑒 𝑣𝑡~𝑁 0,𝑉𝑡

𝛽1,𝑡 = 𝛽1,𝑡−1 + 𝜔1,𝑡 𝑤𝑕𝑒𝑟𝑒 𝜔1,𝑡~𝑁 0,𝑊1,𝑡 (2.3.4)

𝛽2,𝑡 = 𝛽2,𝑡−1 + 𝜔2,𝑡 𝑤𝑕𝑒𝑟𝑒 𝜔2,𝑡~𝑁 0,𝑊2,𝑡

Where yt is the dependent variable, yt-1 is its first lagged term, xt-1 is an additional predictor, and

vt is the observation error term which follows a normal distribution with mean 0, variance Vt. β1,t

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and β2,t are time-varying regression coefficients with evolution terms ω1,t and ω2,t which follow

a normal distribution with zero mean and variance W1,t and W2,t respectively.

The major model assumption is that the observational and the coefficient evolution terms can be

modeled as a normal distribution, which violates the fundamental assumptions behind the

calculations our volatility measures. In the case of realized variance, Anderson, Bollerslev,

Diebold, and Labys (2000a) document that the distribution for realized variance is notably

skewed, but move toward symmetry when we perform a logarithmic transformation on it. Thus,

in our Bayesian analysis, we chose to model the logarithm of RV, and we note that the results

from this model are not directly comparable to those obtained from the HAR models. However,

as with the HAR-RAV analysis, within-model results from the Bayesian approach can still be

used to corroborate the overall findings.

The prior distribution for the coefficient βi is defined as

𝑝 βi,t−1

| 𝐷𝑡−1 ~ 𝑇(𝑚𝑡−1,𝐶𝑡−1) (2.3.5)

Where Dt-1 is the data up to time t-1, mt-1 is the expected value of the coefficient, and Ct-1 is the

variance. Since we must specify an initial uninformative prior, there is a burn-in period for

coefficient estimates before the model stabilizes.

The posterior distribution for the coefficient βi is defined as

𝑝 βi,t

| 𝐷𝑡 ~ 𝑇(𝑚𝑡 ,𝐶𝑡) (2.3.6)

Where the updating equations for mt and Ct are

𝑚𝑡 = 𝑚𝑡−1 + 𝑓 Wt

Vt ∗ (yt − mt−1xt) 2.3.7

𝐶𝑡 = (𝑓 𝑊𝑡 − Vt ) 2.3.8

yt – mt-1xt gives the expected portion of y not explained by x, and f represents a function. Wt is the

underlying variation in the coefficient term and Vt is the observational error; their ratio represents

a measure of how much the change in yt can be attributed to real movements of the coefficient

rather than noise. Overall mt can be thought of simply as mt-1 plus movement in the underlying

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level, and Ct is determined by changes in the relationship between observational and evolution

variances.

3. Data

This dataset consists of 9 foreign exchange rates to the US dollar as well as Brent Crude Oil

futures prices and Comex Gold future prices in US dollars, obtained from the vendor

forextickdata.com, from January 2, 2009 to June 30, 2009, approximately 6 months worth of

data. The data is high-frequency, providing 5-minute prices from 9:35AM to 4:00PM, excluding

weekends. The currencies are listed as follows: AUDUSD, CHFUSD, EURUSD, GBPUSD,

JPYUSD, NZDUSD, CADUSD, NOKUSD, ZARUSD.

Currencies were chosen to be representative of the world markets although consideration was

given to the expected magnitude of the effects of oil and gold on a given currency. Australia,

Switzerland, New Zealand, and South Africa are large gold economies; Norway and Canada are

large oil producers; and the United States is a large importer of both oil and gold. As such, we

expect AUDUSD, CHFUSD, NZDUSD, and ZARUSD follow gold activity while CADUSD and

NOKUSD follow oil activity, on the whole.

Due to the sensitivity of the OLS analysis to outliers, we removed instances of extreme returns,

or jumps, in order to lessen the influence of leverage points in our regressions. Not surprisingly,

these extreme values correspond to unexpected macroeconomic announcements that occur within

the time window.

3.1 Data Assumptions: Market microstructure noise

Although the use of high-frequency data can greatly improve estimates of volatility, it also has

the risk of capturing market microstructure noise, which represents the variation in the asset spot

price from its fundamental value pt . Mathematically this is represented as

𝑝𝑡∗ = 𝑝𝑡 + 𝑒𝑡 (3.1.1)

Where pt* is the spot price and et is the deviation. This inclusion of microstructure noise in prices

can distort the volatility measurements used in our analysis. Anderson, Bollerslev, Diebold, and

Labys (2000b) advocate the use of a volatility signature plot, which presents a graph of the

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average realized variance for each sampling frequency. The smallest sampling interval which

displays a consistent value of average realized variance with those calculated with lower

frequencies is considered optimal. Since the frequencies of our dataset can only be in intervals of

5 minutes, we could not construct a full signature plot; however, the average realized variance

for the 5 minute frequency was relatively consistent to those of the 10 and 15 minute

frequencies, which provide some support for our assumption that 5 minute price data is optimal.

4. Research Methodology

4.1 HAR regressions

Consider the HAR-RV equation (2.3.2).

To determine the possible influential effects of the RV of a commodity on the RV of a particular

currency, we simply added the commodity RV at t-1 into the regression.

𝑅𝑉𝑐𝑢𝑟𝑟 :𝑡+1 = 𝛼 + 𝛽𝑑𝑅𝑉𝑐𝑢𝑟𝑟 :𝑡 + 𝛽𝑤𝑅𝑉𝑐𝑢𝑟𝑟 :𝑡−5,𝑡 + 𝛽𝑚𝑅𝑉𝑐𝑢𝑟𝑟 :𝑡−2,𝑡 + 𝛽𝑐𝑜𝑚𝑚 𝑅𝑉𝑐𝑜𝑚𝑚 :𝑡 + 𝜀𝑡+1 (4.1.1)

This regression is performed for each currency, for each commodity. Table 1 shows the oil RV

coefficient estimate for each currency as well as its individual p-value, while table 3 presents the

gold RV coefficient estimates and p-values. We then compare the statistical values obtained this

model to those obtained the original HAR-RV without the commodity RV to assess significance.

Table 2 and 4 shows a side-by-side comparison of these models for the regressions in which the

commodity RV coefficient was found to be individually significant.

Now consider the HAR-RAV equation (2.3.3).

The analysis for determining significance in commodity RAV follows the same methodology as

described above; we just replace the RVs with their corresponding RAV components. The HAR-

RAV equation with the inclusion of the t-1 commodity RAV is presented below.

𝑅𝐴𝑉𝑐𝑢𝑟𝑟 :𝑡+1 = 𝛼 + 𝛽𝑑𝑅𝐴𝑉𝑐𝑢𝑟𝑟 :𝑡 + 𝛽𝑤𝑅𝐴𝑉𝑐𝑢𝑟𝑟 :𝑡−5,𝑡 + 𝛽𝑚𝑅𝐴𝑉𝑐𝑢𝑟𝑟 :𝑡−22,𝑡 + 𝛽𝑐𝑜𝑚𝑚 𝑅𝑉𝑐𝑜𝑚𝑚 :𝑡 + 𝜀𝑡+1 (4.1.2)

Table 5 shows the oil RAV coefficient estimate for each currency as well as its individual p-

value, while table 7 presents the gold RAV coefficient estimates and p-values. Again, we

compare the statistical values obtained this model to those obtained the original HAR-RAV

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without the commodity RAV to assess significance. Table 6 and 8 shows a side-by-side

comparison of these models for the regressions in which the commodity RV coefficient was

found to be individually significant.

4.2 Bayesian DLM

In the dynamic linear model, we perform a similar analysis, but allow the regression coefficients

to be time-varying. The basic model is as follows:

log(𝑅𝑉𝑐𝑢𝑟𝑟 :𝑡) = 𝛼𝑡 + 𝛽1,𝑡 log(RVcurr :t−1) + 𝛽2,𝑡 log(RVcomm : t−1) + 𝑣𝑡 𝑤𝑕𝑒𝑟𝑒 𝑣𝑡~𝑁 0,𝑉𝑡

𝛽1,𝑡 = 𝛽1,𝑡−1 + 𝜔1,𝑡 𝑤𝑕𝑒𝑟𝑒 𝜔1,𝑡~𝑁 0,𝑊1,𝑡 (4.2.1)

𝛽2,𝑡 = 𝛽2,𝑡−1 + 𝜔2,𝑡 𝑤𝑕𝑒𝑟𝑒 𝜔2,𝑡~𝑁 0,𝑊2,𝑡

Where RVt is the currency realized variance at time t, RVt-1 is the t-1 currency realized variance,

RVcomm:t-1 is the t-1 realized variance of the commodity, and vt is the observation error term

which follows a normal distribution with mean 0, variance Vt. β1,t and β2,t are the time-varying

regression coefficients with evolution terms ω1,t and ω2,t following a normal distribution with

zero mean and variance W1,t and W2,t respectively. Assumptions for the DLM are addressed in

section 2.3.2.

Focusing on the posterior coefficient estimates and 95% credible intervals of the commodity RV

regressor β2,t, we establish significance if its credible interval does not include zero for the

entirety of the sampling period minus the month of January, which will be used for burn-ins.

The Bayesian analysis of RAV was done using the same methodology described above, where

the basic equations are analogous to equation (4.2.1), but uses RAV instead of RV.

log(𝑅𝐴𝑉𝑐𝑢𝑟𝑟 :𝑡) = 𝛼𝑡 + 𝛽1,𝑡 log(RAVcurr :t−1) + 𝛽2,𝑡 log(RAVcomm : t−1) + 𝑣𝑡 𝑤𝑕𝑒𝑟𝑒 𝑣𝑡~𝑁 0,𝑉𝑡

𝛽1,𝑡 = 𝛽1,𝑡−1 + 𝜔1,𝑡 𝑤𝑕𝑒𝑟𝑒 𝜔1,𝑡~𝑁 0,𝑊1,𝑡 (4.2.2)

𝛽2,𝑡 = 𝛽2,𝑡−1 + 𝜔2,𝑡 𝑤𝑕𝑒𝑟𝑒 𝜔2,𝑡~𝑁 0,𝑊2,𝑡

4.3 Additional Remarks

Note that in our analysis, HAR-RV model is in variance terms, HAR-RAV model is in standard

deviation terms, and the Bayesian DLM model is in logarithmic terms. Due to the theoretical

backgrounds and assumptions of each model, as described above, results obtained from these

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models are not directly comparable even if we were to perform transformations to a standardized

term. Instead, the purpose of the analysis is to compare within model significance and see if this

significance is corroborated among the different models.

5. Results

5.1 HAR-RV analysis

Commodity coefficient estimates and their respective p-values for each currency HAR-RV

regression are listed in table 1 and 3.

From table 1, we find the coefficient for oil RV at time t-1 to only be significant in the regression

for CADUSD RV with a p-value of 0.0027. Comparing the relative increase in fit by adding oil

RV to the original HAR-RV for CADUSD, we see a relative large R2 improvement of 0.0923

(see table 2).

From table 3, we find the coefficient for gold RV at time t-1 to be significant in the regressions

for AUDUSD and ZARUSD RVs with the p-values of 0.0379 and 0.0341, respectively.

Comparing the relative increase in fit by adding gold RV to the original HAR-RV model for

these currencies, we see a modest R2 improvement of 0.0372 for AUDUSD and 0.0446 for

ZARUSD (see table 4).

5.2 HAR-RAV analysis

Commodity coefficient estimates and their respective p-values for each currency HAR-RAV

regression are listed in tables 5 and 7.

From table 5, we find the coefficient for oil RAV at time t-1 to be significant in the regressions

for CADUSD and NOKUSD RVs with p-values of 0.0097 and 0.0484, respectively. Comparing

the relative increase in fit by adding oil RV to the original HAR-RAV for CADUSD, we see a R2

improvement of 0.0659 (see table 6). Similarly, the R2 in the model for NOKUSD increased to

0.0351. Note that NOKUSD was not found to be significant at the 5% level in its corresponding

HAR-RV regression.

Looking at table 7, we find the coefficient for gold RAV at time t-1 to be significant in the

AUDUSD, ZARUSD, and NZDUSD RAV regressions. Their corresponding p-values are

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0.0183, 0.0049, and 0.0184, respectively. The overall R2 improvements from adding in the gold

RAV variable in the original HAR-RAV model are as follows: 0.044 for the AUDUSD RAV

regression, 0.0726 for ZARUSD, and 0.0489 for NZDUSD (see table 8). In this case, notice that

gold is significant at the 5% level in the HAR-RAV regression for NZDUSD and not in HAR-

RV model for NZDUSD.

5.3 Bayesian Analysis

Modeling the logarithm of realized variance with the dynamic linear model, we find similar

results to the ones found from the HAR-RV analysis, namely that the oil coefficient was

consistently significant from zero throughout the sampling period for the CADUSD RV

regression, and that the gold coefficient was consistently significant for AUDUSD RV and

ZARUSD RV. A plot of the time-varying posterior coefficient for gold in the AUDUSD RV

regression is presented in figure 1, and a plot of the time-varying posterior coefficient for oil in

the CADUSD RV regression is presented in figure 2. We do not include the month of January in

the plots to account for model burn-in.

Re-running the model using realized absolute variance, we also find matching results to the ones

in obtained in the HAR-RAV model: the time-varying oil coefficient is significant through the

sampling period for the CADUSD and NOKUSD regressions, and the time-varying gold

coefficient is significant for the AUDUSD, ZARUSD, and NOKUSD regressions. The plot of the

gold posterior RAV coefficient for the AUDUSD is presented in figure 3, and the oil posterior

RAV for CADUSD is shown in figure 4. Note that the gold coefficient in the AUDUSD RAV

regression is more or less stable around 0.3 while the oil coefficient in the CADUSD RAV

regression starts out at 0.7 and gradually decline to stabilize around 0.2. The same pattern is seen

in the complementary RV figures, 1 and 2.

Conclusion

Following previous work done on relationship between the commodity and foreign exchange

markets, we extend the analysis to examine the link between the volatility of the two asset

classes using the two high-frequency volatility measures, realized variance and realized absolute

value. In particular, we are interested in the ability of a commodity’s volatility to predict the

volatility of a currency, especially those whose economies are largely dependent on that

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particular commodity. We expect CADUSD and NOKUSD to follow oil more closely than other

currencies and AUDUSD, CHFUSD, NZDUSD, and ZARUSD to be more related to gold than

others (see section 3). After conducting comprehensive analysis in the HAR framework with

both RV and RAV as well as performing corresponding analysis using a Bayesian DLM

perspective, we find corroborating evidence among all of our models that oil volatility can be a

useful predictor for CADUSD volatility and that gold volatility can be a useful predictor for the

AUDUSD and ZARUSD volatilities. In our HAR and DLM models using realized absolute

variance, we also find indications that oil volatility can be predictive of the volatility of

NOKUSD and that gold volatility can be useful in NZDUSD volatility predictions although this

is not seen in models using realized variance.

Diebold and Yilman (2010) suggest that these instances of volatility spillovers can be attributed

to general uncertainty caused by a global financial crisis and the onset of herd mentality. Indeed,

they found that the overall spillover index in the markets rose to over thirty percent during the

first half of 2009 as the effects of the financial crisis rippled through the world economies. Our

findings are not inconsistent with these results. However, since our research only uses data from

the first half of 2009, we can only conclude that there is evidence of significance in the

commodity volatility regressors to predict volatility in foreign exchange for that time period. We

are unable generalized our work to determine if this significance can be attributed to increased

uncertainty in a financial crisis, an overall long-term relationship between these currency-

commodity pairs, or a combination of the two effects.

Further research can be done to explore this question by rerunning our analysis with a dataset

consisting of more currency-pairs for a larger sampling period as well as incorporating the

measurement of volatility spillover detailed in Diebold and Yilman (2009).

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𝑅𝑉𝑐𝑢𝑟𝑟 : 𝑡+1 = 𝛼 + 𝛽𝑑𝑅𝑉𝑐𝑢𝑟𝑟 :𝑡 + 𝛽𝑤𝑅𝑉𝑐𝑢𝑟𝑟 :𝑡−5,𝑡 + 𝛽𝑚𝑅𝑉𝑐𝑢𝑟𝑟 :𝑡−22,𝑡 + 𝛽𝑜𝑖𝑙𝑅𝑉𝑜𝑖𝑙 ,𝑡 + 𝜀𝑡+1

Table 1: HAR-RV regressions: Oil RV coefficient and coefficient p-values for each currency regression

CADUSD

CADUSD w/ RVoil

α 0.0002

(0.0001) 0.0001

(0.0001)

β1 0.0316

(0.1012)

-0.0375

(0.0994)

β2 0.0145

(0.1016) 0.0122

(0.0973)

β3 0.0038

(0.1027) -0.0044

(0.0984)

β4 -

0.3240**

(0.1034)

P-value of F-

Test 0.9896 0.0935

R2

0.0012 0.0935

*indicates significance at 5% level, ** significant at 1% level

Table 2: HAR-RV regressions: Model comparison (with standard deviations) between original HAR-RV

model and the model with the oil RV for CADUSD

AUDUSD CHFUSD EURUSD GBPUSD JPYUSD

β4 0.0228 0.0165 0.0087 0.0124 0.0249

β4

P-value 0.2560 0.4898 0.3459 0.3289 0.3751

NZDUSD CADUSD NOKUSD ZARUSD

β4 0.0177 0.3240 0.0341 0.0056

β4

P-value 0.3342 0.0027 0.1420 0.5788

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𝑅𝑉𝑐𝑢𝑟𝑟 : 𝑡+1 = 𝛼 + 𝛽𝑑𝑅𝑉𝑐𝑢𝑟𝑟 :𝑡 + 𝛽𝑤𝑅𝑉𝑐𝑢𝑟𝑟 :𝑡−5,𝑡 + 𝛽𝑚𝑅𝑉𝑐𝑢𝑟𝑟 :𝑡−22,𝑡 + 𝛽𝑔𝑜𝑙𝑑 𝑅𝑉𝑔𝑜𝑙𝑑 ,𝑡 + 𝜀𝑡+1

Table 3: HAR-RV regressions: Gold RV coefficient and coefficient p-values for each currency regression

AUDUSD

AUDUSD w/ RVgold

ZARUSD ZARUSD w/

RVgold

α 0.0000

(0.0000) 0.0000

(0.0000) 0.0000

(0.0000) 0.0000

(0.0000)

β1 0.4137** (0.0914)

0.4011** (0.0900)

0.1867 (0.0992)

0.1494 (0.0988)

β2 -0.0537 (0.0927)

-0.0421 (0.0912)

-0.1108 (0.0974)

-0.0877 (0.0962)

β3 -0.0244 (0.1091)

-0.0662 (0.1090)

0.0769 (0.0911)

0.0461 (0.0905)

β4 - 0.0647* (0.0303)

- 0.0367* (0.0168)

P-value of F-Test

0.0003 0.0001 0.1588 0.0442

R2 0.1754 0.2126 0.0523 0.0969

*indicates significance at 5% level, ** significant at 1% level

Table 4: HAR-RV regressions: Model comparison (with standard deviations) between original HAR-RV

model and the model with the gold RV for AUDUSD and ZARUSD

AUDUSD CHFUSD EURUSD GBPUSD JPYUSD

βgold 0.0647 0.0444 0.0283 0.0356 0.0178

βgold P-value

0.0379 0.2854 0.0762 0.0933 0.0928

NZDUSD CADUSD NOKUSD ZARUSD

βgold 0.0560 0.2252 0.0763 0.0367

βgold P-value

0.0628 0.2494 0.0687 0.0341

15

𝑅𝐴𝑉𝑐𝑢𝑟𝑟 : 𝑡+1 = 𝛼 + 𝛽𝑑𝑅𝐴𝑉𝑐𝑢𝑟𝑟 :𝑡 + 𝛽𝑤𝑅𝐴𝑉𝑐𝑢𝑟𝑟 :𝑡−5,𝑡 + 𝛽𝑚𝑅𝐴𝑉𝑐𝑢𝑟𝑟 :𝑡−22,𝑡 + 𝛽𝑜𝑖𝑙𝑅𝐴𝑉𝑜𝑖𝑙 ,𝑡 + 𝜀𝑡+1

Table 5: HAR-RAV regressions: Oil RAV coefficient and coefficient p-values for each currency

regression

CADUSD

CADUSD w/ RAVoil

NOKUSD NOKUSD w/ RAVoil

α 0.0105

(0.0025) 0.0096

(0.0024) 0.0047

(0.0012) 0.0050

(0.0012)

β1 0.1637

(0.1003) 0.0623

(0.1045) 0.3234*

(0.0991) 0.2611*

(0.1024)

β2 0.0446

(0.1013) 0.0056

(0.0994) 0.0722

(0.0998) 0.0310

(0.1004)

β3 0.0581

(0.1037) 0.0265

(0.1014) 0.0324

(0.0848) -0.0424

(0.0914)

β4 -

0.1856**

(0.0698) -

0.0662*

(0.0329)

P-value of F-

Test 0.3359 0.0379 0.0037 0.0016

R2

0.0345 0.1004 0.1307 0.1658

*indicates significance at 5% level, ** significant at 1% level

Table 6: HAR-RAV regressions: Model comparison between original HAR-RAV model and the model

with the oil RAV for CADUSD and NOKUSD

AUDUSD CHFUSD EURUSD GBPUSD JPYUSD

β4 0.0392 0.0284 0.0230 0.0480 0.0588

β4

P-value 0.2272 0.2623 0.2710 0.0816 0.1597

NZDUSD CADUSD NOKUSD ZARUSD

β4 0.0629 0.1856 0.0662 0.0163

β4

P-value 0.1529 0.0097 0.0484 0.4400

16

𝑅𝐴𝑉𝑐𝑢𝑟𝑟 : 𝑡+1 = 𝛼 + 𝛽𝑑𝑅𝐴𝑉𝑐𝑢𝑟𝑟 :𝑡 + 𝛽𝑤𝑅𝐴𝑉𝑐𝑢𝑟𝑟 :𝑡−5,𝑡 + 𝛽𝑚𝑅𝐴𝑉𝑐𝑢𝑟𝑟 :𝑡−22,𝑡 + 𝛽𝑔𝑜𝑙𝑑 𝑅𝐴𝑉𝑔𝑜𝑙𝑑 ,𝑡 + 𝜀𝑡+1

Table 7: HAR-RAV regressions: Gold RAV coefficient and coefficient p-values for each currency

regression

AUDUSD

AUDUSD w/ RAVgold

ZARUSD ZARUSD w/

RAVgold NZDUSD

NZDUSD w/ RAVgold

α 0.0042

(0.0010) 0.0041

(0.0010) 0.0055

(0.0010) 0.0053

(0.0009)

0.0060

(0.0013) 0.0060

(0.0012)

β1 0.4708** (0.0894)

0.4257** (0.0892)

0.3122** (0.0983)

0.2670** (0.0961)

0.3722**

(0.0938) 0.3190**

(0.0942)

β2 -0.0361 (0.0943)

-0.0286 (0.0921)

-0.0928 (0.0961)

-0.0597 (0.0934)

-0.0476

(0.0943) -0.0132

(0.0932)

β3 0.0247

(0.0867) -0.0495 (0.0901)

-0.0375 (0.0878)

-0.1243 (0.0898)

-0.0267

(0.0902) -0.1324

(0.0983)

β4 - 0.1198* (0.0495)

- 0.1056** (0.0364)

- 0.1326*

(0.0549)

P-value of F-Test

0.0000 0.0000 0.0221 0.0015 0.0019 0.0004

R2 0.2315 0.2755 0.0949 0.1675 0.1429 0.1918

*indicates significance at 5% level, ** significant at 1% level

Table 8: HAR-RAV regressions: Model comparison between original HAR-RAV model and the model

with the gold RAV for AUDUSD, ZARUSD, and NZDUSD

AUDUSD CHFUSD EURUSD GBPUSD JPYUSD

βgold 0.1198 0.1172 0.0936 0.1403 0.1571

βgold P-value

0.0183 0.0782 0.0618 0.0658 0.0809

NZDUSD CADUSD NOKUSD ZARUSD

βgold 0.1326 0.3728 0.1845 0.1056

βgold P-value

0.0184 0.1216 0.0724 0.0049

17

Bayesian Dynamic Linear Model Posterior Coefficient Plots with 95% Credible Intervals

Figure 1 (above left): Graph of the time-varying RV gold coefficient when regressing for AUDUSD Figure 2 (above right): Graph of the time-varying RV oil coefficient when regressing for CADUSD

Figure 3 (above left): Graph of the time-varying RAV gold coefficient when regressing for AUDUSD Figure 4 (above right): Graph of the time-varying RAV oil coefficient when regressing for CADUSD

18

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